Properties

Label 8281.2.a.cd.1.3
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.90903\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.264627 q^{2} +2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} +1.03998 q^{8} +5.46247 q^{9} +O(q^{10})\) \(q-0.264627 q^{2} +2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} +1.03998 q^{8} +5.46247 q^{9} -0.379780 q^{10} -5.50474 q^{11} -5.61435 q^{12} +4.17491 q^{15} +3.58474 q^{16} +4.83072 q^{17} -1.44552 q^{18} -2.82036 q^{19} -2.76981 q^{20} +1.45670 q^{22} -5.99956 q^{23} +3.02532 q^{24} -2.94033 q^{25} +7.16341 q^{27} +1.04188 q^{29} -1.10479 q^{30} -9.20895 q^{31} -3.02857 q^{32} -16.0135 q^{33} -1.27834 q^{34} -10.5424 q^{36} -0.612497 q^{37} +0.746342 q^{38} +1.49252 q^{40} +10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} +7.83949 q^{45} +1.58764 q^{46} -2.40922 q^{47} +10.4281 q^{48} +0.778091 q^{50} +14.0527 q^{51} -1.82959 q^{53} -1.89563 q^{54} -7.90015 q^{55} -8.20452 q^{57} -0.275709 q^{58} +0.870914 q^{59} -8.05746 q^{60} -3.33253 q^{61} +2.43693 q^{62} -6.36804 q^{64} +4.23759 q^{66} +6.62741 q^{67} -9.32316 q^{68} -17.4529 q^{69} +6.85856 q^{71} +5.68083 q^{72} -3.14147 q^{73} +0.162083 q^{74} -8.55353 q^{75} +5.44322 q^{76} -17.5723 q^{79} +5.14465 q^{80} +4.45118 q^{81} -2.81022 q^{82} -11.4525 q^{83} +6.93283 q^{85} +2.23261 q^{86} +3.03086 q^{87} -5.72479 q^{88} +0.995318 q^{89} -2.07454 q^{90} +11.5790 q^{92} -26.7891 q^{93} +0.637545 q^{94} -4.04765 q^{95} -8.81020 q^{96} -13.5090 q^{97} -30.0695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + 4q^{10} - 4q^{11} - 4q^{12} - 12q^{15} + 16q^{17} + 4q^{18} - 2q^{19} - 16q^{20} - 12q^{22} - 6q^{23} - 12q^{24} - 4q^{25} + 20q^{27} - 6q^{29} - 6q^{31} + 20q^{32} - 4q^{33} - 24q^{36} + 8q^{38} + 4q^{40} + 8q^{41} + 2q^{43} + 4q^{44} - 14q^{45} - 8q^{46} - 30q^{47} - 8q^{48} - 8q^{50} - 4q^{51} - 14q^{53} + 48q^{54} - 8q^{55} - 4q^{57} + 8q^{58} - 24q^{59} - 12q^{60} + 28q^{62} - 20q^{64} - 4q^{66} - 16q^{67} + 28q^{68} - 20q^{69} - 8q^{71} - 28q^{72} + 6q^{73} - 12q^{74} + 12q^{75} + 16q^{76} - 22q^{79} + 28q^{80} + 46q^{81} - 40q^{82} - 50q^{83} + 8q^{85} + 16q^{86} - 16q^{87} - 44q^{88} - 26q^{89} - 40q^{90} + 20q^{92} - 16q^{93} - 32q^{94} - 6q^{95} + 20q^{96} + 14q^{97} - 12q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.264627 −0.187119 −0.0935596 0.995614i \(-0.529825\pi\)
−0.0935596 + 0.995614i \(0.529825\pi\)
\(3\) 2.90903 1.67953 0.839765 0.542949i \(-0.182692\pi\)
0.839765 + 0.542949i \(0.182692\pi\)
\(4\) −1.92997 −0.964986
\(5\) 1.43515 0.641820 0.320910 0.947110i \(-0.396011\pi\)
0.320910 + 0.947110i \(0.396011\pi\)
\(6\) −0.769807 −0.314273
\(7\) 0 0
\(8\) 1.03998 0.367687
\(9\) 5.46247 1.82082
\(10\) −0.379780 −0.120097
\(11\) −5.50474 −1.65974 −0.829871 0.557955i \(-0.811586\pi\)
−0.829871 + 0.557955i \(0.811586\pi\)
\(12\) −5.61435 −1.62072
\(13\) 0 0
\(14\) 0 0
\(15\) 4.17491 1.07796
\(16\) 3.58474 0.896185
\(17\) 4.83072 1.17162 0.585811 0.810448i \(-0.300776\pi\)
0.585811 + 0.810448i \(0.300776\pi\)
\(18\) −1.44552 −0.340711
\(19\) −2.82036 −0.647035 −0.323518 0.946222i \(-0.604865\pi\)
−0.323518 + 0.946222i \(0.604865\pi\)
\(20\) −2.76981 −0.619348
\(21\) 0 0
\(22\) 1.45670 0.310570
\(23\) −5.99956 −1.25100 −0.625498 0.780226i \(-0.715104\pi\)
−0.625498 + 0.780226i \(0.715104\pi\)
\(24\) 3.02532 0.617541
\(25\) −2.94033 −0.588067
\(26\) 0 0
\(27\) 7.16341 1.37860
\(28\) 0 0
\(29\) 1.04188 0.193472 0.0967361 0.995310i \(-0.469160\pi\)
0.0967361 + 0.995310i \(0.469160\pi\)
\(30\) −1.10479 −0.201706
\(31\) −9.20895 −1.65398 −0.826988 0.562219i \(-0.809948\pi\)
−0.826988 + 0.562219i \(0.809948\pi\)
\(32\) −3.02857 −0.535380
\(33\) −16.0135 −2.78759
\(34\) −1.27834 −0.219233
\(35\) 0 0
\(36\) −10.5424 −1.75707
\(37\) −0.612497 −0.100694 −0.0503470 0.998732i \(-0.516033\pi\)
−0.0503470 + 0.998732i \(0.516033\pi\)
\(38\) 0.746342 0.121073
\(39\) 0 0
\(40\) 1.49252 0.235989
\(41\) 10.6196 1.65850 0.829249 0.558879i \(-0.188768\pi\)
0.829249 + 0.558879i \(0.188768\pi\)
\(42\) 0 0
\(43\) −8.43685 −1.28661 −0.643304 0.765611i \(-0.722437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(44\) 10.6240 1.60163
\(45\) 7.83949 1.16864
\(46\) 1.58764 0.234085
\(47\) −2.40922 −0.351422 −0.175711 0.984442i \(-0.556222\pi\)
−0.175711 + 0.984442i \(0.556222\pi\)
\(48\) 10.4281 1.50517
\(49\) 0 0
\(50\) 0.778091 0.110039
\(51\) 14.0527 1.96778
\(52\) 0 0
\(53\) −1.82959 −0.251313 −0.125657 0.992074i \(-0.540104\pi\)
−0.125657 + 0.992074i \(0.540104\pi\)
\(54\) −1.89563 −0.257962
\(55\) −7.90015 −1.06526
\(56\) 0 0
\(57\) −8.20452 −1.08672
\(58\) −0.275709 −0.0362024
\(59\) 0.870914 0.113383 0.0566917 0.998392i \(-0.481945\pi\)
0.0566917 + 0.998392i \(0.481945\pi\)
\(60\) −8.05746 −1.04021
\(61\) −3.33253 −0.426686 −0.213343 0.976977i \(-0.568435\pi\)
−0.213343 + 0.976977i \(0.568435\pi\)
\(62\) 2.43693 0.309491
\(63\) 0 0
\(64\) −6.36804 −0.796005
\(65\) 0 0
\(66\) 4.23759 0.521611
\(67\) 6.62741 0.809667 0.404833 0.914390i \(-0.367329\pi\)
0.404833 + 0.914390i \(0.367329\pi\)
\(68\) −9.32316 −1.13060
\(69\) −17.4529 −2.10108
\(70\) 0 0
\(71\) 6.85856 0.813961 0.406980 0.913437i \(-0.366582\pi\)
0.406980 + 0.913437i \(0.366582\pi\)
\(72\) 5.68083 0.669493
\(73\) −3.14147 −0.367682 −0.183841 0.982956i \(-0.558853\pi\)
−0.183841 + 0.982956i \(0.558853\pi\)
\(74\) 0.162083 0.0188418
\(75\) −8.55353 −0.987676
\(76\) 5.44322 0.624380
\(77\) 0 0
\(78\) 0 0
\(79\) −17.5723 −1.97704 −0.988518 0.151101i \(-0.951718\pi\)
−0.988518 + 0.151101i \(0.951718\pi\)
\(80\) 5.14465 0.575190
\(81\) 4.45118 0.494576
\(82\) −2.81022 −0.310337
\(83\) −11.4525 −1.25708 −0.628538 0.777779i \(-0.716346\pi\)
−0.628538 + 0.777779i \(0.716346\pi\)
\(84\) 0 0
\(85\) 6.93283 0.751971
\(86\) 2.23261 0.240749
\(87\) 3.03086 0.324943
\(88\) −5.72479 −0.610265
\(89\) 0.995318 0.105503 0.0527517 0.998608i \(-0.483201\pi\)
0.0527517 + 0.998608i \(0.483201\pi\)
\(90\) −2.07454 −0.218675
\(91\) 0 0
\(92\) 11.5790 1.20719
\(93\) −26.7891 −2.77790
\(94\) 0.637545 0.0657577
\(95\) −4.04765 −0.415280
\(96\) −8.81020 −0.899188
\(97\) −13.5090 −1.37163 −0.685817 0.727774i \(-0.740555\pi\)
−0.685817 + 0.727774i \(0.740555\pi\)
\(98\) 0 0
\(99\) −30.0695 −3.02210
\(100\) 5.67477 0.567477
\(101\) −1.00807 −0.100306 −0.0501532 0.998742i \(-0.515971\pi\)
−0.0501532 + 0.998742i \(0.515971\pi\)
\(102\) −3.71873 −0.368209
\(103\) −12.7754 −1.25880 −0.629401 0.777081i \(-0.716700\pi\)
−0.629401 + 0.777081i \(0.716700\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.484157 0.0470255
\(107\) −0.685495 −0.0662693 −0.0331347 0.999451i \(-0.510549\pi\)
−0.0331347 + 0.999451i \(0.510549\pi\)
\(108\) −13.8252 −1.33033
\(109\) 2.90344 0.278099 0.139050 0.990285i \(-0.455595\pi\)
0.139050 + 0.990285i \(0.455595\pi\)
\(110\) 2.09059 0.199330
\(111\) −1.78177 −0.169119
\(112\) 0 0
\(113\) 12.0315 1.13183 0.565915 0.824464i \(-0.308523\pi\)
0.565915 + 0.824464i \(0.308523\pi\)
\(114\) 2.17113 0.203345
\(115\) −8.61029 −0.802914
\(116\) −2.01080 −0.186698
\(117\) 0 0
\(118\) −0.230467 −0.0212162
\(119\) 0 0
\(120\) 4.34180 0.396350
\(121\) 19.3022 1.75474
\(122\) 0.881875 0.0798412
\(123\) 30.8927 2.78550
\(124\) 17.7730 1.59606
\(125\) −11.3956 −1.01925
\(126\) 0 0
\(127\) 15.6659 1.39012 0.695062 0.718950i \(-0.255377\pi\)
0.695062 + 0.718950i \(0.255377\pi\)
\(128\) 7.74229 0.684328
\(129\) −24.5431 −2.16090
\(130\) 0 0
\(131\) 12.1273 1.05957 0.529784 0.848132i \(-0.322273\pi\)
0.529784 + 0.848132i \(0.322273\pi\)
\(132\) 30.9056 2.68998
\(133\) 0 0
\(134\) −1.75379 −0.151504
\(135\) 10.2806 0.884813
\(136\) 5.02383 0.430790
\(137\) −15.9375 −1.36163 −0.680815 0.732456i \(-0.738374\pi\)
−0.680815 + 0.732456i \(0.738374\pi\)
\(138\) 4.61851 0.393153
\(139\) −6.64088 −0.563272 −0.281636 0.959521i \(-0.590877\pi\)
−0.281636 + 0.959521i \(0.590877\pi\)
\(140\) 0 0
\(141\) −7.00851 −0.590223
\(142\) −1.81496 −0.152308
\(143\) 0 0
\(144\) 19.5815 1.63180
\(145\) 1.49526 0.124174
\(146\) 0.831317 0.0688003
\(147\) 0 0
\(148\) 1.18210 0.0971683
\(149\) 19.5502 1.60162 0.800809 0.598920i \(-0.204403\pi\)
0.800809 + 0.598920i \(0.204403\pi\)
\(150\) 2.26349 0.184813
\(151\) −10.6880 −0.869779 −0.434890 0.900484i \(-0.643213\pi\)
−0.434890 + 0.900484i \(0.643213\pi\)
\(152\) −2.93311 −0.237906
\(153\) 26.3877 2.13332
\(154\) 0 0
\(155\) −13.2163 −1.06156
\(156\) 0 0
\(157\) −15.0734 −1.20299 −0.601496 0.798876i \(-0.705428\pi\)
−0.601496 + 0.798876i \(0.705428\pi\)
\(158\) 4.65009 0.369942
\(159\) −5.32233 −0.422088
\(160\) −4.34646 −0.343618
\(161\) 0 0
\(162\) −1.17790 −0.0925447
\(163\) 23.8135 1.86521 0.932607 0.360894i \(-0.117528\pi\)
0.932607 + 0.360894i \(0.117528\pi\)
\(164\) −20.4955 −1.60043
\(165\) −22.9818 −1.78913
\(166\) 3.03064 0.235223
\(167\) 7.12371 0.551249 0.275625 0.961265i \(-0.411115\pi\)
0.275625 + 0.961265i \(0.411115\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1.83461 −0.140708
\(171\) −15.4061 −1.17814
\(172\) 16.2829 1.24156
\(173\) 11.2367 0.854309 0.427155 0.904179i \(-0.359516\pi\)
0.427155 + 0.904179i \(0.359516\pi\)
\(174\) −0.802047 −0.0608030
\(175\) 0 0
\(176\) −19.7331 −1.48744
\(177\) 2.53352 0.190431
\(178\) −0.263387 −0.0197417
\(179\) −13.1945 −0.986204 −0.493102 0.869972i \(-0.664137\pi\)
−0.493102 + 0.869972i \(0.664137\pi\)
\(180\) −15.1300 −1.12772
\(181\) −13.7414 −1.02139 −0.510696 0.859761i \(-0.670612\pi\)
−0.510696 + 0.859761i \(0.670612\pi\)
\(182\) 0 0
\(183\) −9.69443 −0.716633
\(184\) −6.23939 −0.459974
\(185\) −0.879028 −0.0646274
\(186\) 7.08912 0.519799
\(187\) −26.5919 −1.94459
\(188\) 4.64974 0.339117
\(189\) 0 0
\(190\) 1.07112 0.0777069
\(191\) −16.3307 −1.18165 −0.590824 0.806800i \(-0.701197\pi\)
−0.590824 + 0.806800i \(0.701197\pi\)
\(192\) −18.5248 −1.33692
\(193\) 14.0533 1.01158 0.505790 0.862656i \(-0.331201\pi\)
0.505790 + 0.862656i \(0.331201\pi\)
\(194\) 3.57485 0.256659
\(195\) 0 0
\(196\) 0 0
\(197\) 1.46898 0.104660 0.0523302 0.998630i \(-0.483335\pi\)
0.0523302 + 0.998630i \(0.483335\pi\)
\(198\) 7.95719 0.565493
\(199\) −13.3772 −0.948285 −0.474142 0.880448i \(-0.657242\pi\)
−0.474142 + 0.880448i \(0.657242\pi\)
\(200\) −3.05787 −0.216224
\(201\) 19.2794 1.35986
\(202\) 0.266761 0.0187692
\(203\) 0 0
\(204\) −27.1214 −1.89888
\(205\) 15.2407 1.06446
\(206\) 3.38072 0.235546
\(207\) −32.7724 −2.27784
\(208\) 0 0
\(209\) 15.5254 1.07391
\(210\) 0 0
\(211\) 3.47044 0.238915 0.119457 0.992839i \(-0.461885\pi\)
0.119457 + 0.992839i \(0.461885\pi\)
\(212\) 3.53105 0.242514
\(213\) 19.9518 1.36707
\(214\) 0.181400 0.0124003
\(215\) −12.1082 −0.825771
\(216\) 7.44977 0.506893
\(217\) 0 0
\(218\) −0.768328 −0.0520378
\(219\) −9.13865 −0.617533
\(220\) 15.2471 1.02796
\(221\) 0 0
\(222\) 0.471505 0.0316453
\(223\) 9.91318 0.663836 0.331918 0.943308i \(-0.392304\pi\)
0.331918 + 0.943308i \(0.392304\pi\)
\(224\) 0 0
\(225\) −16.0615 −1.07077
\(226\) −3.18386 −0.211787
\(227\) −12.0727 −0.801292 −0.400646 0.916233i \(-0.631214\pi\)
−0.400646 + 0.916233i \(0.631214\pi\)
\(228\) 15.8345 1.04867
\(229\) −4.05171 −0.267745 −0.133872 0.990999i \(-0.542741\pi\)
−0.133872 + 0.990999i \(0.542741\pi\)
\(230\) 2.27851 0.150241
\(231\) 0 0
\(232\) 1.08353 0.0711372
\(233\) 12.5450 0.821850 0.410925 0.911669i \(-0.365206\pi\)
0.410925 + 0.911669i \(0.365206\pi\)
\(234\) 0 0
\(235\) −3.45761 −0.225549
\(236\) −1.68084 −0.109413
\(237\) −51.1184 −3.32049
\(238\) 0 0
\(239\) −13.3463 −0.863299 −0.431649 0.902042i \(-0.642068\pi\)
−0.431649 + 0.902042i \(0.642068\pi\)
\(240\) 14.9660 0.966049
\(241\) −20.3854 −1.31314 −0.656568 0.754267i \(-0.727993\pi\)
−0.656568 + 0.754267i \(0.727993\pi\)
\(242\) −5.10787 −0.328346
\(243\) −8.54160 −0.547944
\(244\) 6.43169 0.411747
\(245\) 0 0
\(246\) −8.17502 −0.521221
\(247\) 0 0
\(248\) −9.57708 −0.608145
\(249\) −33.3157 −2.11130
\(250\) 3.01558 0.190722
\(251\) 17.1921 1.08515 0.542577 0.840006i \(-0.317449\pi\)
0.542577 + 0.840006i \(0.317449\pi\)
\(252\) 0 0
\(253\) 33.0260 2.07633
\(254\) −4.14561 −0.260119
\(255\) 20.1678 1.26296
\(256\) 10.6873 0.667954
\(257\) −7.64695 −0.477004 −0.238502 0.971142i \(-0.576656\pi\)
−0.238502 + 0.971142i \(0.576656\pi\)
\(258\) 6.49475 0.404345
\(259\) 0 0
\(260\) 0 0
\(261\) 5.69124 0.352279
\(262\) −3.20921 −0.198266
\(263\) −0.101037 −0.00623022 −0.00311511 0.999995i \(-0.500992\pi\)
−0.00311511 + 0.999995i \(0.500992\pi\)
\(264\) −16.6536 −1.02496
\(265\) −2.62574 −0.161298
\(266\) 0 0
\(267\) 2.89541 0.177196
\(268\) −12.7907 −0.781318
\(269\) −7.56852 −0.461461 −0.230730 0.973018i \(-0.574111\pi\)
−0.230730 + 0.973018i \(0.574111\pi\)
\(270\) −2.72052 −0.165565
\(271\) 13.8554 0.841653 0.420826 0.907141i \(-0.361740\pi\)
0.420826 + 0.907141i \(0.361740\pi\)
\(272\) 17.3169 1.04999
\(273\) 0 0
\(274\) 4.21748 0.254787
\(275\) 16.1858 0.976039
\(276\) 33.6837 2.02752
\(277\) 0.552935 0.0332226 0.0166113 0.999862i \(-0.494712\pi\)
0.0166113 + 0.999862i \(0.494712\pi\)
\(278\) 1.75735 0.105399
\(279\) −50.3036 −3.01160
\(280\) 0 0
\(281\) −1.14667 −0.0684043 −0.0342022 0.999415i \(-0.510889\pi\)
−0.0342022 + 0.999415i \(0.510889\pi\)
\(282\) 1.85464 0.110442
\(283\) −4.05396 −0.240983 −0.120491 0.992714i \(-0.538447\pi\)
−0.120491 + 0.992714i \(0.538447\pi\)
\(284\) −13.2368 −0.785461
\(285\) −11.7747 −0.697476
\(286\) 0 0
\(287\) 0 0
\(288\) −16.5435 −0.974833
\(289\) 6.33588 0.372699
\(290\) −0.395685 −0.0232354
\(291\) −39.2982 −2.30370
\(292\) 6.06296 0.354808
\(293\) −15.0649 −0.880102 −0.440051 0.897973i \(-0.645040\pi\)
−0.440051 + 0.897973i \(0.645040\pi\)
\(294\) 0 0
\(295\) 1.24990 0.0727718
\(296\) −0.636982 −0.0370238
\(297\) −39.4327 −2.28812
\(298\) −5.17351 −0.299694
\(299\) 0 0
\(300\) 16.5081 0.953094
\(301\) 0 0
\(302\) 2.82834 0.162752
\(303\) −2.93250 −0.168468
\(304\) −10.1103 −0.579863
\(305\) −4.78269 −0.273856
\(306\) −6.98288 −0.399185
\(307\) 19.9408 1.13808 0.569040 0.822310i \(-0.307315\pi\)
0.569040 + 0.822310i \(0.307315\pi\)
\(308\) 0 0
\(309\) −37.1642 −2.11420
\(310\) 3.49737 0.198637
\(311\) −10.8956 −0.617833 −0.308916 0.951089i \(-0.599966\pi\)
−0.308916 + 0.951089i \(0.599966\pi\)
\(312\) 0 0
\(313\) −0.0519190 −0.00293464 −0.00146732 0.999999i \(-0.500467\pi\)
−0.00146732 + 0.999999i \(0.500467\pi\)
\(314\) 3.98883 0.225103
\(315\) 0 0
\(316\) 33.9140 1.90781
\(317\) −16.1010 −0.904321 −0.452161 0.891937i \(-0.649347\pi\)
−0.452161 + 0.891937i \(0.649347\pi\)
\(318\) 1.40843 0.0789808
\(319\) −5.73528 −0.321114
\(320\) −9.13912 −0.510892
\(321\) −1.99413 −0.111301
\(322\) 0 0
\(323\) −13.6244 −0.758081
\(324\) −8.59066 −0.477259
\(325\) 0 0
\(326\) −6.30167 −0.349017
\(327\) 8.44621 0.467077
\(328\) 11.0441 0.609808
\(329\) 0 0
\(330\) 6.08159 0.334781
\(331\) −30.7862 −1.69216 −0.846081 0.533054i \(-0.821044\pi\)
−0.846081 + 0.533054i \(0.821044\pi\)
\(332\) 22.1030 1.21306
\(333\) −3.34575 −0.183346
\(334\) −1.88512 −0.103149
\(335\) 9.51135 0.519661
\(336\) 0 0
\(337\) −2.41842 −0.131740 −0.0658700 0.997828i \(-0.520982\pi\)
−0.0658700 + 0.997828i \(0.520982\pi\)
\(338\) 0 0
\(339\) 35.0001 1.90094
\(340\) −13.3802 −0.725642
\(341\) 50.6929 2.74517
\(342\) 4.07687 0.220452
\(343\) 0 0
\(344\) −8.77411 −0.473069
\(345\) −25.0476 −1.34852
\(346\) −2.97353 −0.159858
\(347\) −0.492527 −0.0264403 −0.0132201 0.999913i \(-0.504208\pi\)
−0.0132201 + 0.999913i \(0.504208\pi\)
\(348\) −5.84948 −0.313565
\(349\) 11.9442 0.639356 0.319678 0.947526i \(-0.396425\pi\)
0.319678 + 0.947526i \(0.396425\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.6715 0.888593
\(353\) −15.5299 −0.826575 −0.413288 0.910601i \(-0.635620\pi\)
−0.413288 + 0.910601i \(0.635620\pi\)
\(354\) −0.670436 −0.0356333
\(355\) 9.84308 0.522417
\(356\) −1.92094 −0.101809
\(357\) 0 0
\(358\) 3.49162 0.184538
\(359\) 8.50709 0.448987 0.224493 0.974476i \(-0.427927\pi\)
0.224493 + 0.974476i \(0.427927\pi\)
\(360\) 8.15287 0.429694
\(361\) −11.0456 −0.581345
\(362\) 3.63635 0.191122
\(363\) 56.1507 2.94715
\(364\) 0 0
\(365\) −4.50850 −0.235985
\(366\) 2.56540 0.134096
\(367\) 5.19084 0.270960 0.135480 0.990780i \(-0.456742\pi\)
0.135480 + 0.990780i \(0.456742\pi\)
\(368\) −21.5069 −1.12112
\(369\) 58.0091 3.01983
\(370\) 0.232614 0.0120930
\(371\) 0 0
\(372\) 51.7023 2.68064
\(373\) −10.1427 −0.525169 −0.262585 0.964909i \(-0.584575\pi\)
−0.262585 + 0.964909i \(0.584575\pi\)
\(374\) 7.03692 0.363870
\(375\) −33.1502 −1.71187
\(376\) −2.50553 −0.129213
\(377\) 0 0
\(378\) 0 0
\(379\) 3.63670 0.186805 0.0934024 0.995628i \(-0.470226\pi\)
0.0934024 + 0.995628i \(0.470226\pi\)
\(380\) 7.81186 0.400740
\(381\) 45.5726 2.33476
\(382\) 4.32154 0.221109
\(383\) −4.60281 −0.235192 −0.117596 0.993061i \(-0.537519\pi\)
−0.117596 + 0.993061i \(0.537519\pi\)
\(384\) 22.5226 1.14935
\(385\) 0 0
\(386\) −3.71888 −0.189286
\(387\) −46.0861 −2.34269
\(388\) 26.0721 1.32361
\(389\) 19.6104 0.994286 0.497143 0.867669i \(-0.334382\pi\)
0.497143 + 0.867669i \(0.334382\pi\)
\(390\) 0 0
\(391\) −28.9822 −1.46569
\(392\) 0 0
\(393\) 35.2788 1.77958
\(394\) −0.388731 −0.0195840
\(395\) −25.2189 −1.26890
\(396\) 58.0333 2.91628
\(397\) 19.8635 0.996919 0.498459 0.866913i \(-0.333899\pi\)
0.498459 + 0.866913i \(0.333899\pi\)
\(398\) 3.53996 0.177442
\(399\) 0 0
\(400\) −10.5403 −0.527017
\(401\) 15.1117 0.754644 0.377322 0.926082i \(-0.376845\pi\)
0.377322 + 0.926082i \(0.376845\pi\)
\(402\) −5.10183 −0.254456
\(403\) 0 0
\(404\) 1.94554 0.0967943
\(405\) 6.38813 0.317429
\(406\) 0 0
\(407\) 3.37164 0.167126
\(408\) 14.6145 0.723525
\(409\) 35.2443 1.74272 0.871360 0.490644i \(-0.163238\pi\)
0.871360 + 0.490644i \(0.163238\pi\)
\(410\) −4.03310 −0.199181
\(411\) −46.3626 −2.28690
\(412\) 24.6563 1.21473
\(413\) 0 0
\(414\) 8.67246 0.426228
\(415\) −16.4361 −0.806816
\(416\) 0 0
\(417\) −19.3185 −0.946033
\(418\) −4.10842 −0.200950
\(419\) 1.50468 0.0735084 0.0367542 0.999324i \(-0.488298\pi\)
0.0367542 + 0.999324i \(0.488298\pi\)
\(420\) 0 0
\(421\) 24.5079 1.19444 0.597221 0.802077i \(-0.296272\pi\)
0.597221 + 0.802077i \(0.296272\pi\)
\(422\) −0.918370 −0.0447056
\(423\) −13.1603 −0.639877
\(424\) −1.90272 −0.0924045
\(425\) −14.2039 −0.688992
\(426\) −5.27977 −0.255806
\(427\) 0 0
\(428\) 1.32299 0.0639490
\(429\) 0 0
\(430\) 3.20414 0.154518
\(431\) −41.0655 −1.97805 −0.989027 0.147732i \(-0.952803\pi\)
−0.989027 + 0.147732i \(0.952803\pi\)
\(432\) 25.6790 1.23548
\(433\) 6.65603 0.319869 0.159934 0.987128i \(-0.448872\pi\)
0.159934 + 0.987128i \(0.448872\pi\)
\(434\) 0 0
\(435\) 4.34975 0.208555
\(436\) −5.60357 −0.268362
\(437\) 16.9209 0.809438
\(438\) 2.41833 0.115552
\(439\) 8.22990 0.392792 0.196396 0.980525i \(-0.437076\pi\)
0.196396 + 0.980525i \(0.437076\pi\)
\(440\) −8.21596 −0.391681
\(441\) 0 0
\(442\) 0 0
\(443\) 17.6856 0.840266 0.420133 0.907463i \(-0.361983\pi\)
0.420133 + 0.907463i \(0.361983\pi\)
\(444\) 3.43878 0.163197
\(445\) 1.42843 0.0677143
\(446\) −2.62329 −0.124216
\(447\) 56.8723 2.68997
\(448\) 0 0
\(449\) 14.5250 0.685477 0.342738 0.939431i \(-0.388646\pi\)
0.342738 + 0.939431i \(0.388646\pi\)
\(450\) 4.25030 0.200361
\(451\) −58.4580 −2.75268
\(452\) −23.2205 −1.09220
\(453\) −31.0918 −1.46082
\(454\) 3.19475 0.149937
\(455\) 0 0
\(456\) −8.53250 −0.399571
\(457\) −3.78919 −0.177251 −0.0886255 0.996065i \(-0.528247\pi\)
−0.0886255 + 0.996065i \(0.528247\pi\)
\(458\) 1.07219 0.0501002
\(459\) 34.6045 1.61520
\(460\) 16.6176 0.774801
\(461\) −13.1107 −0.610627 −0.305314 0.952252i \(-0.598761\pi\)
−0.305314 + 0.952252i \(0.598761\pi\)
\(462\) 0 0
\(463\) −15.3027 −0.711176 −0.355588 0.934643i \(-0.615719\pi\)
−0.355588 + 0.934643i \(0.615719\pi\)
\(464\) 3.73487 0.173387
\(465\) −38.4465 −1.78292
\(466\) −3.31974 −0.153784
\(467\) 30.7738 1.42404 0.712022 0.702158i \(-0.247780\pi\)
0.712022 + 0.702158i \(0.247780\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.914975 0.0422046
\(471\) −43.8491 −2.02046
\(472\) 0.905729 0.0416896
\(473\) 46.4427 2.13544
\(474\) 13.5273 0.621328
\(475\) 8.29280 0.380500
\(476\) 0 0
\(477\) −9.99407 −0.457597
\(478\) 3.53178 0.161540
\(479\) 1.71740 0.0784699 0.0392350 0.999230i \(-0.487508\pi\)
0.0392350 + 0.999230i \(0.487508\pi\)
\(480\) −12.6440 −0.577117
\(481\) 0 0
\(482\) 5.39451 0.245713
\(483\) 0 0
\(484\) −37.2527 −1.69330
\(485\) −19.3875 −0.880342
\(486\) 2.26033 0.102531
\(487\) −22.6805 −1.02775 −0.513877 0.857864i \(-0.671791\pi\)
−0.513877 + 0.857864i \(0.671791\pi\)
\(488\) −3.46575 −0.156887
\(489\) 69.2741 3.13268
\(490\) 0 0
\(491\) −13.1366 −0.592846 −0.296423 0.955057i \(-0.595794\pi\)
−0.296423 + 0.955057i \(0.595794\pi\)
\(492\) −59.6220 −2.68797
\(493\) 5.03303 0.226676
\(494\) 0 0
\(495\) −43.1543 −1.93964
\(496\) −33.0117 −1.48227
\(497\) 0 0
\(498\) 8.81622 0.395064
\(499\) −14.0395 −0.628495 −0.314248 0.949341i \(-0.601752\pi\)
−0.314248 + 0.949341i \(0.601752\pi\)
\(500\) 21.9932 0.983566
\(501\) 20.7231 0.925840
\(502\) −4.54948 −0.203053
\(503\) 0.367865 0.0164023 0.00820114 0.999966i \(-0.497389\pi\)
0.00820114 + 0.999966i \(0.497389\pi\)
\(504\) 0 0
\(505\) −1.44673 −0.0643786
\(506\) −8.73957 −0.388521
\(507\) 0 0
\(508\) −30.2348 −1.34145
\(509\) −41.2319 −1.82757 −0.913787 0.406194i \(-0.866856\pi\)
−0.913787 + 0.406194i \(0.866856\pi\)
\(510\) −5.33694 −0.236324
\(511\) 0 0
\(512\) −18.3127 −0.809315
\(513\) −20.2034 −0.892002
\(514\) 2.02359 0.0892566
\(515\) −18.3347 −0.807925
\(516\) 47.3675 2.08524
\(517\) 13.2622 0.583269
\(518\) 0 0
\(519\) 32.6879 1.43484
\(520\) 0 0
\(521\) 1.04099 0.0456065 0.0228032 0.999740i \(-0.492741\pi\)
0.0228032 + 0.999740i \(0.492741\pi\)
\(522\) −1.50605 −0.0659182
\(523\) −20.0209 −0.875451 −0.437726 0.899109i \(-0.644216\pi\)
−0.437726 + 0.899109i \(0.644216\pi\)
\(524\) −23.4054 −1.02247
\(525\) 0 0
\(526\) 0.0267371 0.00116579
\(527\) −44.4859 −1.93784
\(528\) −57.4042 −2.49820
\(529\) 12.9947 0.564989
\(530\) 0.694840 0.0301819
\(531\) 4.75735 0.206451
\(532\) 0 0
\(533\) 0 0
\(534\) −0.766203 −0.0331568
\(535\) −0.983791 −0.0425330
\(536\) 6.89234 0.297704
\(537\) −38.3833 −1.65636
\(538\) 2.00283 0.0863481
\(539\) 0 0
\(540\) −19.8413 −0.853832
\(541\) 9.78749 0.420797 0.210399 0.977616i \(-0.432524\pi\)
0.210399 + 0.977616i \(0.432524\pi\)
\(542\) −3.66649 −0.157489
\(543\) −39.9743 −1.71546
\(544\) −14.6302 −0.627263
\(545\) 4.16689 0.178490
\(546\) 0 0
\(547\) −2.56174 −0.109532 −0.0547660 0.998499i \(-0.517441\pi\)
−0.0547660 + 0.998499i \(0.517441\pi\)
\(548\) 30.7589 1.31395
\(549\) −18.2038 −0.776921
\(550\) −4.28319 −0.182636
\(551\) −2.93848 −0.125183
\(552\) −18.1506 −0.772541
\(553\) 0 0
\(554\) −0.146321 −0.00621660
\(555\) −2.55712 −0.108544
\(556\) 12.8167 0.543550
\(557\) 27.4442 1.16285 0.581424 0.813601i \(-0.302496\pi\)
0.581424 + 0.813601i \(0.302496\pi\)
\(558\) 13.3117 0.563528
\(559\) 0 0
\(560\) 0 0
\(561\) −77.3567 −3.26600
\(562\) 0.303438 0.0127998
\(563\) 0.162708 0.00685734 0.00342867 0.999994i \(-0.498909\pi\)
0.00342867 + 0.999994i \(0.498909\pi\)
\(564\) 13.5262 0.569557
\(565\) 17.2671 0.726431
\(566\) 1.07278 0.0450925
\(567\) 0 0
\(568\) 7.13273 0.299283
\(569\) −12.3901 −0.519419 −0.259709 0.965687i \(-0.583627\pi\)
−0.259709 + 0.965687i \(0.583627\pi\)
\(570\) 3.11591 0.130511
\(571\) −21.8122 −0.912810 −0.456405 0.889772i \(-0.650863\pi\)
−0.456405 + 0.889772i \(0.650863\pi\)
\(572\) 0 0
\(573\) −47.5066 −1.98462
\(574\) 0 0
\(575\) 17.6407 0.735669
\(576\) −34.7852 −1.44939
\(577\) −6.06583 −0.252524 −0.126262 0.991997i \(-0.540298\pi\)
−0.126262 + 0.991997i \(0.540298\pi\)
\(578\) −1.67664 −0.0697391
\(579\) 40.8816 1.69898
\(580\) −2.88581 −0.119827
\(581\) 0 0
\(582\) 10.3993 0.431067
\(583\) 10.0714 0.417115
\(584\) −3.26705 −0.135192
\(585\) 0 0
\(586\) 3.98658 0.164684
\(587\) −20.5820 −0.849510 −0.424755 0.905308i \(-0.639640\pi\)
−0.424755 + 0.905308i \(0.639640\pi\)
\(588\) 0 0
\(589\) 25.9726 1.07018
\(590\) −0.330756 −0.0136170
\(591\) 4.27331 0.175781
\(592\) −2.19564 −0.0902404
\(593\) 24.0397 0.987190 0.493595 0.869692i \(-0.335682\pi\)
0.493595 + 0.869692i \(0.335682\pi\)
\(594\) 10.4349 0.428151
\(595\) 0 0
\(596\) −37.7314 −1.54554
\(597\) −38.9147 −1.59267
\(598\) 0 0
\(599\) 32.2523 1.31779 0.658896 0.752234i \(-0.271024\pi\)
0.658896 + 0.752234i \(0.271024\pi\)
\(600\) −8.89546 −0.363156
\(601\) 5.21454 0.212705 0.106353 0.994328i \(-0.466083\pi\)
0.106353 + 0.994328i \(0.466083\pi\)
\(602\) 0 0
\(603\) 36.2020 1.47426
\(604\) 20.6276 0.839325
\(605\) 27.7016 1.12623
\(606\) 0.776017 0.0315235
\(607\) 9.07048 0.368160 0.184080 0.982911i \(-0.441070\pi\)
0.184080 + 0.982911i \(0.441070\pi\)
\(608\) 8.54165 0.346410
\(609\) 0 0
\(610\) 1.26563 0.0512437
\(611\) 0 0
\(612\) −50.9275 −2.05862
\(613\) −20.0920 −0.811507 −0.405754 0.913983i \(-0.632991\pi\)
−0.405754 + 0.913983i \(0.632991\pi\)
\(614\) −5.27686 −0.212957
\(615\) 44.3357 1.78779
\(616\) 0 0
\(617\) 12.9556 0.521572 0.260786 0.965397i \(-0.416018\pi\)
0.260786 + 0.965397i \(0.416018\pi\)
\(618\) 9.83463 0.395607
\(619\) 44.3644 1.78316 0.891578 0.452866i \(-0.149598\pi\)
0.891578 + 0.452866i \(0.149598\pi\)
\(620\) 25.5070 1.02439
\(621\) −42.9773 −1.72462
\(622\) 2.88327 0.115608
\(623\) 0 0
\(624\) 0 0
\(625\) −1.65276 −0.0661105
\(626\) 0.0137392 0.000549127 0
\(627\) 45.1638 1.80367
\(628\) 29.0913 1.16087
\(629\) −2.95880 −0.117975
\(630\) 0 0
\(631\) −6.61717 −0.263426 −0.131713 0.991288i \(-0.542048\pi\)
−0.131713 + 0.991288i \(0.542048\pi\)
\(632\) −18.2747 −0.726930
\(633\) 10.0956 0.401265
\(634\) 4.26075 0.169216
\(635\) 22.4830 0.892210
\(636\) 10.2719 0.407309
\(637\) 0 0
\(638\) 1.51771 0.0600866
\(639\) 37.4647 1.48208
\(640\) 11.1114 0.439216
\(641\) 18.9567 0.748744 0.374372 0.927279i \(-0.377858\pi\)
0.374372 + 0.927279i \(0.377858\pi\)
\(642\) 0.527699 0.0208266
\(643\) −13.4019 −0.528517 −0.264259 0.964452i \(-0.585127\pi\)
−0.264259 + 0.964452i \(0.585127\pi\)
\(644\) 0 0
\(645\) −35.2231 −1.38691
\(646\) 3.60537 0.141851
\(647\) 42.7588 1.68102 0.840511 0.541794i \(-0.182255\pi\)
0.840511 + 0.541794i \(0.182255\pi\)
\(648\) 4.62912 0.181849
\(649\) −4.79416 −0.188187
\(650\) 0 0
\(651\) 0 0
\(652\) −45.9593 −1.79991
\(653\) 10.9852 0.429884 0.214942 0.976627i \(-0.431044\pi\)
0.214942 + 0.976627i \(0.431044\pi\)
\(654\) −2.23509 −0.0873990
\(655\) 17.4046 0.680052
\(656\) 38.0684 1.48632
\(657\) −17.1602 −0.669483
\(658\) 0 0
\(659\) −17.7614 −0.691884 −0.345942 0.938256i \(-0.612441\pi\)
−0.345942 + 0.938256i \(0.612441\pi\)
\(660\) 44.3542 1.72649
\(661\) −8.18255 −0.318264 −0.159132 0.987257i \(-0.550870\pi\)
−0.159132 + 0.987257i \(0.550870\pi\)
\(662\) 8.14685 0.316636
\(663\) 0 0
\(664\) −11.9103 −0.462210
\(665\) 0 0
\(666\) 0.885374 0.0343076
\(667\) −6.25082 −0.242033
\(668\) −13.7486 −0.531948
\(669\) 28.8378 1.11493
\(670\) −2.51696 −0.0972385
\(671\) 18.3447 0.708189
\(672\) 0 0
\(673\) 9.30129 0.358539 0.179269 0.983800i \(-0.442627\pi\)
0.179269 + 0.983800i \(0.442627\pi\)
\(674\) 0.639979 0.0246511
\(675\) −21.0628 −0.810708
\(676\) 0 0
\(677\) −41.1552 −1.58172 −0.790862 0.611994i \(-0.790367\pi\)
−0.790862 + 0.611994i \(0.790367\pi\)
\(678\) −9.26195 −0.355703
\(679\) 0 0
\(680\) 7.20997 0.276490
\(681\) −35.1198 −1.34579
\(682\) −13.4147 −0.513675
\(683\) −39.2842 −1.50317 −0.751583 0.659638i \(-0.770709\pi\)
−0.751583 + 0.659638i \(0.770709\pi\)
\(684\) 29.7334 1.13689
\(685\) −22.8727 −0.873921
\(686\) 0 0
\(687\) −11.7866 −0.449685
\(688\) −30.2439 −1.15304
\(689\) 0 0
\(690\) 6.62827 0.252334
\(691\) 3.03355 0.115402 0.0577009 0.998334i \(-0.481623\pi\)
0.0577009 + 0.998334i \(0.481623\pi\)
\(692\) −21.6865 −0.824397
\(693\) 0 0
\(694\) 0.130336 0.00494748
\(695\) −9.53069 −0.361520
\(696\) 3.15202 0.119477
\(697\) 51.3002 1.94313
\(698\) −3.16074 −0.119636
\(699\) 36.4938 1.38032
\(700\) 0 0
\(701\) −26.2320 −0.990767 −0.495384 0.868674i \(-0.664972\pi\)
−0.495384 + 0.868674i \(0.664972\pi\)
\(702\) 0 0
\(703\) 1.72746 0.0651525
\(704\) 35.0544 1.32116
\(705\) −10.0583 −0.378817
\(706\) 4.10963 0.154668
\(707\) 0 0
\(708\) −4.88962 −0.183763
\(709\) 7.87770 0.295853 0.147927 0.988998i \(-0.452740\pi\)
0.147927 + 0.988998i \(0.452740\pi\)
\(710\) −2.60474 −0.0977542
\(711\) −95.9881 −3.59984
\(712\) 1.03511 0.0387922
\(713\) 55.2497 2.06912
\(714\) 0 0
\(715\) 0 0
\(716\) 25.4650 0.951673
\(717\) −38.8247 −1.44994
\(718\) −2.25120 −0.0840141
\(719\) 45.6656 1.70304 0.851519 0.524323i \(-0.175682\pi\)
0.851519 + 0.524323i \(0.175682\pi\)
\(720\) 28.1025 1.04732
\(721\) 0 0
\(722\) 2.92295 0.108781
\(723\) −59.3017 −2.20545
\(724\) 26.5206 0.985630
\(725\) −3.06348 −0.113775
\(726\) −14.8590 −0.551468
\(727\) 37.5947 1.39431 0.697155 0.716921i \(-0.254449\pi\)
0.697155 + 0.716921i \(0.254449\pi\)
\(728\) 0 0
\(729\) −38.2013 −1.41486
\(730\) 1.19307 0.0441574
\(731\) −40.7561 −1.50742
\(732\) 18.7100 0.691541
\(733\) −53.1810 −1.96429 −0.982143 0.188138i \(-0.939755\pi\)
−0.982143 + 0.188138i \(0.939755\pi\)
\(734\) −1.37363 −0.0507018
\(735\) 0 0
\(736\) 18.1701 0.669758
\(737\) −36.4822 −1.34384
\(738\) −15.3508 −0.565069
\(739\) −41.9633 −1.54364 −0.771822 0.635839i \(-0.780654\pi\)
−0.771822 + 0.635839i \(0.780654\pi\)
\(740\) 1.69650 0.0623646
\(741\) 0 0
\(742\) 0 0
\(743\) 38.5424 1.41398 0.706991 0.707222i \(-0.250052\pi\)
0.706991 + 0.707222i \(0.250052\pi\)
\(744\) −27.8600 −1.02140
\(745\) 28.0576 1.02795
\(746\) 2.68403 0.0982693
\(747\) −62.5590 −2.28891
\(748\) 51.3216 1.87650
\(749\) 0 0
\(750\) 8.77242 0.320323
\(751\) 36.2434 1.32254 0.661270 0.750148i \(-0.270018\pi\)
0.661270 + 0.750148i \(0.270018\pi\)
\(752\) −8.63644 −0.314939
\(753\) 50.0123 1.82255
\(754\) 0 0
\(755\) −15.3390 −0.558242
\(756\) 0 0
\(757\) 19.4752 0.707837 0.353919 0.935276i \(-0.384849\pi\)
0.353919 + 0.935276i \(0.384849\pi\)
\(758\) −0.962368 −0.0349548
\(759\) 96.0738 3.48726
\(760\) −4.20946 −0.152693
\(761\) 51.9059 1.88159 0.940793 0.338981i \(-0.110082\pi\)
0.940793 + 0.338981i \(0.110082\pi\)
\(762\) −12.0597 −0.436878
\(763\) 0 0
\(764\) 31.5178 1.14028
\(765\) 37.8704 1.36921
\(766\) 1.21803 0.0440090
\(767\) 0 0
\(768\) 31.0896 1.12185
\(769\) 7.31376 0.263741 0.131870 0.991267i \(-0.457902\pi\)
0.131870 + 0.991267i \(0.457902\pi\)
\(770\) 0 0
\(771\) −22.2452 −0.801142
\(772\) −27.1225 −0.976161
\(773\) 14.1844 0.510178 0.255089 0.966918i \(-0.417895\pi\)
0.255089 + 0.966918i \(0.417895\pi\)
\(774\) 12.1956 0.438362
\(775\) 27.0774 0.972649
\(776\) −14.0491 −0.504332
\(777\) 0 0
\(778\) −5.18943 −0.186050
\(779\) −29.9510 −1.07311
\(780\) 0 0
\(781\) −37.7546 −1.35097
\(782\) 7.66946 0.274259
\(783\) 7.46341 0.266721
\(784\) 0 0
\(785\) −21.6327 −0.772104
\(786\) −9.33569 −0.332993
\(787\) −31.2777 −1.11493 −0.557465 0.830201i \(-0.688226\pi\)
−0.557465 + 0.830201i \(0.688226\pi\)
\(788\) −2.83509 −0.100996
\(789\) −0.293921 −0.0104639
\(790\) 6.67360 0.237436
\(791\) 0 0
\(792\) −31.2715 −1.11119
\(793\) 0 0
\(794\) −5.25640 −0.186543
\(795\) −7.63836 −0.270905
\(796\) 25.8176 0.915082
\(797\) 20.2422 0.717017 0.358509 0.933526i \(-0.383285\pi\)
0.358509 + 0.933526i \(0.383285\pi\)
\(798\) 0 0
\(799\) −11.6383 −0.411733
\(800\) 8.90500 0.314839
\(801\) 5.43689 0.192103
\(802\) −3.99897 −0.141208
\(803\) 17.2930 0.610257
\(804\) −37.2086 −1.31225
\(805\) 0 0
\(806\) 0 0
\(807\) −22.0171 −0.775037
\(808\) −1.04836 −0.0368813
\(809\) 7.88265 0.277139 0.138570 0.990353i \(-0.455750\pi\)
0.138570 + 0.990353i \(0.455750\pi\)
\(810\) −1.69047 −0.0593970
\(811\) 5.99962 0.210675 0.105338 0.994437i \(-0.466408\pi\)
0.105338 + 0.994437i \(0.466408\pi\)
\(812\) 0 0
\(813\) 40.3057 1.41358
\(814\) −0.892226 −0.0312725
\(815\) 34.1760 1.19713
\(816\) 50.3754 1.76349
\(817\) 23.7950 0.832480
\(818\) −9.32659 −0.326096
\(819\) 0 0
\(820\) −29.4142 −1.02719
\(821\) 19.1692 0.669011 0.334505 0.942394i \(-0.391431\pi\)
0.334505 + 0.942394i \(0.391431\pi\)
\(822\) 12.2688 0.427923
\(823\) −30.3735 −1.05875 −0.529376 0.848387i \(-0.677574\pi\)
−0.529376 + 0.848387i \(0.677574\pi\)
\(824\) −13.2861 −0.462845
\(825\) 47.0850 1.63929
\(826\) 0 0
\(827\) 14.6870 0.510717 0.255359 0.966846i \(-0.417807\pi\)
0.255359 + 0.966846i \(0.417807\pi\)
\(828\) 63.2499 2.19809
\(829\) −34.9985 −1.21555 −0.607774 0.794110i \(-0.707938\pi\)
−0.607774 + 0.794110i \(0.707938\pi\)
\(830\) 4.34943 0.150971
\(831\) 1.60851 0.0557985
\(832\) 0 0
\(833\) 0 0
\(834\) 5.11220 0.177021
\(835\) 10.2236 0.353803
\(836\) −29.9635 −1.03631
\(837\) −65.9675 −2.28017
\(838\) −0.398178 −0.0137548
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) 0 0
\(841\) −27.9145 −0.962568
\(842\) −6.48544 −0.223503
\(843\) −3.33569 −0.114887
\(844\) −6.69785 −0.230550
\(845\) 0 0
\(846\) 3.48257 0.119733
\(847\) 0 0
\(848\) −6.55859 −0.225223
\(849\) −11.7931 −0.404738
\(850\) 3.75874 0.128924
\(851\) 3.67472 0.125968
\(852\) −38.5064 −1.31921
\(853\) −32.6336 −1.11735 −0.558676 0.829386i \(-0.688690\pi\)
−0.558676 + 0.829386i \(0.688690\pi\)
\(854\) 0 0
\(855\) −22.1102 −0.756152
\(856\) −0.712898 −0.0243664
\(857\) −18.8742 −0.644730 −0.322365 0.946616i \(-0.604478\pi\)
−0.322365 + 0.946616i \(0.604478\pi\)
\(858\) 0 0
\(859\) 15.8242 0.539915 0.269957 0.962872i \(-0.412990\pi\)
0.269957 + 0.962872i \(0.412990\pi\)
\(860\) 23.3684 0.796857
\(861\) 0 0
\(862\) 10.8670 0.370132
\(863\) −52.3212 −1.78104 −0.890518 0.454948i \(-0.849658\pi\)
−0.890518 + 0.454948i \(0.849658\pi\)
\(864\) −21.6949 −0.738075
\(865\) 16.1264 0.548313
\(866\) −1.76136 −0.0598536
\(867\) 18.4313 0.625959
\(868\) 0 0
\(869\) 96.7309 3.28137
\(870\) −1.15106 −0.0390246
\(871\) 0 0
\(872\) 3.01951 0.102253
\(873\) −73.7927 −2.49750
\(874\) −4.47773 −0.151461
\(875\) 0 0
\(876\) 17.6373 0.595911
\(877\) 54.0162 1.82400 0.911999 0.410193i \(-0.134539\pi\)
0.911999 + 0.410193i \(0.134539\pi\)
\(878\) −2.17785 −0.0734989
\(879\) −43.8243 −1.47816
\(880\) −28.3200 −0.954667
\(881\) −42.0823 −1.41779 −0.708895 0.705314i \(-0.750806\pi\)
−0.708895 + 0.705314i \(0.750806\pi\)
\(882\) 0 0
\(883\) 36.5314 1.22938 0.614689 0.788769i \(-0.289281\pi\)
0.614689 + 0.788769i \(0.289281\pi\)
\(884\) 0 0
\(885\) 3.63599 0.122222
\(886\) −4.68007 −0.157230
\(887\) −11.4648 −0.384950 −0.192475 0.981302i \(-0.561651\pi\)
−0.192475 + 0.981302i \(0.561651\pi\)
\(888\) −1.85300 −0.0621827
\(889\) 0 0
\(890\) −0.378001 −0.0126706
\(891\) −24.5026 −0.820869
\(892\) −19.1322 −0.640592
\(893\) 6.79488 0.227382
\(894\) −15.0499 −0.503345
\(895\) −18.9361 −0.632965
\(896\) 0 0
\(897\) 0 0
\(898\) −3.84370 −0.128266
\(899\) −9.59462 −0.319999
\(900\) 30.9982 1.03327
\(901\) −8.83823 −0.294444
\(902\) 15.4695 0.515079
\(903\) 0 0
\(904\) 12.5125 0.416159
\(905\) −19.7211 −0.655550
\(906\) 8.22772 0.273348
\(907\) −5.04665 −0.167571 −0.0837856 0.996484i \(-0.526701\pi\)
−0.0837856 + 0.996484i \(0.526701\pi\)
\(908\) 23.2999 0.773236
\(909\) −5.50653 −0.182640
\(910\) 0 0
\(911\) 47.5236 1.57453 0.787263 0.616618i \(-0.211497\pi\)
0.787263 + 0.616618i \(0.211497\pi\)
\(912\) −29.4111 −0.973898
\(913\) 63.0431 2.08642
\(914\) 1.00272 0.0331671
\(915\) −13.9130 −0.459950
\(916\) 7.81969 0.258370
\(917\) 0 0
\(918\) −9.15726 −0.302235
\(919\) −45.6698 −1.50651 −0.753254 0.657730i \(-0.771517\pi\)
−0.753254 + 0.657730i \(0.771517\pi\)
\(920\) −8.95449 −0.295221
\(921\) 58.0084 1.91144
\(922\) 3.46945 0.114260
\(923\) 0 0
\(924\) 0 0
\(925\) 1.80095 0.0592148
\(926\) 4.04950 0.133075
\(927\) −69.7855 −2.29206
\(928\) −3.15540 −0.103581
\(929\) 44.4449 1.45819 0.729095 0.684412i \(-0.239941\pi\)
0.729095 + 0.684412i \(0.239941\pi\)
\(930\) 10.1740 0.333618
\(931\) 0 0
\(932\) −24.2115 −0.793074
\(933\) −31.6957 −1.03767
\(934\) −8.14357 −0.266466
\(935\) −38.1634 −1.24808
\(936\) 0 0
\(937\) 46.9796 1.53476 0.767379 0.641194i \(-0.221561\pi\)
0.767379 + 0.641194i \(0.221561\pi\)
\(938\) 0 0
\(939\) −0.151034 −0.00492881
\(940\) 6.67309 0.217652
\(941\) −35.5654 −1.15940 −0.579699 0.814830i \(-0.696830\pi\)
−0.579699 + 0.814830i \(0.696830\pi\)
\(942\) 11.6036 0.378067
\(943\) −63.7128 −2.07477
\(944\) 3.12200 0.101613
\(945\) 0 0
\(946\) −12.2900 −0.399581
\(947\) 22.3592 0.726576 0.363288 0.931677i \(-0.381654\pi\)
0.363288 + 0.931677i \(0.381654\pi\)
\(948\) 98.6570 3.20423
\(949\) 0 0
\(950\) −2.19450 −0.0711989
\(951\) −46.8383 −1.51884
\(952\) 0 0
\(953\) −46.7684 −1.51498 −0.757488 0.652849i \(-0.773574\pi\)
−0.757488 + 0.652849i \(0.773574\pi\)
\(954\) 2.64470 0.0856252
\(955\) −23.4371 −0.758406
\(956\) 25.7579 0.833071
\(957\) −16.6841 −0.539321
\(958\) −0.454469 −0.0146832
\(959\) 0 0
\(960\) −26.5860 −0.858059
\(961\) 53.8048 1.73564
\(962\) 0 0
\(963\) −3.74450 −0.120665
\(964\) 39.3432 1.26716
\(965\) 20.1687 0.649253
\(966\) 0 0
\(967\) −8.22976 −0.264651 −0.132326 0.991206i \(-0.542244\pi\)
−0.132326 + 0.991206i \(0.542244\pi\)
\(968\) 20.0738 0.645196
\(969\) −39.6338 −1.27322
\(970\) 5.13046 0.164729
\(971\) −23.6326 −0.758407 −0.379204 0.925313i \(-0.623802\pi\)
−0.379204 + 0.925313i \(0.623802\pi\)
\(972\) 16.4850 0.528758
\(973\) 0 0
\(974\) 6.00187 0.192312
\(975\) 0 0
\(976\) −11.9462 −0.382390
\(977\) 26.6428 0.852379 0.426190 0.904634i \(-0.359856\pi\)
0.426190 + 0.904634i \(0.359856\pi\)
\(978\) −18.3318 −0.586185
\(979\) −5.47897 −0.175109
\(980\) 0 0
\(981\) 15.8600 0.506370
\(982\) 3.47629 0.110933
\(983\) −43.6302 −1.39159 −0.695793 0.718242i \(-0.744947\pi\)
−0.695793 + 0.718242i \(0.744947\pi\)
\(984\) 32.1276 1.02419
\(985\) 2.10821 0.0671732
\(986\) −1.33187 −0.0424155
\(987\) 0 0
\(988\) 0 0
\(989\) 50.6174 1.60954
\(990\) 11.4198 0.362945
\(991\) −7.60816 −0.241681 −0.120841 0.992672i \(-0.538559\pi\)
−0.120841 + 0.992672i \(0.538559\pi\)
\(992\) 27.8899 0.885506
\(993\) −89.5581 −2.84204
\(994\) 0 0
\(995\) −19.1983 −0.608628
\(996\) 64.2984 2.03737
\(997\) 19.4356 0.615532 0.307766 0.951462i \(-0.400419\pi\)
0.307766 + 0.951462i \(0.400419\pi\)
\(998\) 3.71523 0.117604
\(999\) −4.38757 −0.138817
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8281.2.a.cd.1.3 6
7.6 odd 2 8281.2.a.cc.1.3 6
13.12 even 2 637.2.a.n.1.4 yes 6
39.38 odd 2 5733.2.a.br.1.3 6
91.12 odd 6 637.2.e.o.508.3 12
91.25 even 6 637.2.e.n.79.3 12
91.38 odd 6 637.2.e.o.79.3 12
91.51 even 6 637.2.e.n.508.3 12
91.90 odd 2 637.2.a.m.1.4 6
273.272 even 2 5733.2.a.bu.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.4 6 91.90 odd 2
637.2.a.n.1.4 yes 6 13.12 even 2
637.2.e.n.79.3 12 91.25 even 6
637.2.e.n.508.3 12 91.51 even 6
637.2.e.o.79.3 12 91.38 odd 6
637.2.e.o.508.3 12 91.12 odd 6
5733.2.a.br.1.3 6 39.38 odd 2
5733.2.a.bu.1.3 6 273.272 even 2
8281.2.a.cc.1.3 6 7.6 odd 2
8281.2.a.cd.1.3 6 1.1 even 1 trivial